EFFECT OF AXIAL STIFFENERS ON THE 

BUCKLING PROPERTIES OF THIN CURVED 

PLYWOOD PLATES IN AXIAL COMPRESSION 

March 1948 

This Report is One of a Series 

Issued In Cooperation with the 

ARMY-NAVY-CIVIL COMMITTEE 

on 

AIRCRAFT DESIGN CRITERIA 

Under the Supervision of the 

AERONAUTlCAL BOARD 

No. 1567 

UNITED STATES DEPARTMENT OF AGRICULTURE 

FOREST SERVICE 

FOREST PRODUCTS LABORATORY 

Madison 5, Wisconsin 

In Cooperation with the University of Wisconsin 
EFFECT OF AXIAL STIFFENERS ON THE BUCKLING 

PROPERTIES OF THIN CURVED PLYWOOD PLATES IN 

AXIAL COMPRESSION 

By 

FRED WERREN, Engineer 

and 

C. B. NORRIS, Engineer 

Summary 

This report presents the results of tests and analysis of data on 78 curved 

plywood plates tested in axial compression. Each late was unstiffened or had 

a single stiffener parallel to the applied stress.2 

The results of the tests show that the addition of an axial stiffener to a 

curved plywood plate in compression does not increase the buckling stress of 

the plywood. The effect of the stiffener is to carry additional load and, if 

stiff enough, to break up the buckle pattern. The theoretical buckling stress 

can be determined by the same methods used for a cylinder of finite length.3 

Introduction 

In many of the structural uses of plywood, particularly in aircraft, it is 

possible to provide additional strength and rigidity to a plywood shell by

the addition of stiffeners. The behavior of a specific section of a structure can be approximated by testing a prototype, If a sufficiently rigid 

stiffener is added to a plywood plate, the plywood can be made to buckle on 

either side of the stiffener instead of across the stiffener. 

1This progress report is one of a series prepared and distributed by the 

Forest Products Laboratory under U. S. Navy, Bureau of Aeronautics No. NBA- PO-NAer 00565 and ArmyAirForce No. AAF-PO-(33-038)46-1189E. Results here 

reported are preliminary and may be revised as additional data become 

available. 

2A report of the effect of circumferential stiffeners on curved plywood plates

in axial compression is now under preparation at the Forest Products Laboratory by T. B. Heebink. 3Forest Products Laboratory Report No. 1514. “Effect of Length on the Buckling 

Stress of Thin-walled Plywood Cylinders in Axial Compression,” by E. W. Kuenzi. (Revised 1948) , 

Rept. No. 1567 -1-
It was the purpose of the experiments conducted for this report to determine 

the effect of an axial stiffener on the buckling characteristics of a curved 

plywood plate in axial compression. (Similar teste have been made on plywood cylinders.4) A comparison of the observed buckling stresses of the plates 

and theoretical buckling stresses of comparable cylinders is presented in this 

report. Variables in the curved plywood plates include (1) three constructions, 

(2) two veneer thicknesses, (3) length to width ratios from 1.0 to 3.7, (4)

length to radius ratios from 1.2 to 1.8, (5) width to radius ratios from 0.25 

to 1.0, and (6) grain direction of face plies either parallel or perpendicular 

to the direction of the applied stress. 

Description of Material 

The veneers used in making the plywood plates for these tests were rotary-cut 

yellow birch and yellow-poplar of aircraft grade. Each plate was made from 

veneers of a single species into three-, four-, or five-ply constructions, 

The grain direction of each ply was perpendicular to the grain of the adjacent plies, except in the four-ply construction, in which the grain directions Of 

the two center plies were parallel to each other, The plates were fabricated 

by bag-molding to the desired radius of curvature, using a phenolic-resin film 

glue to bond adjacent plies, 

Axial stiffeners were cut from a single Sitka spruce plank, quarter-sawn into 

strips 5/16 inch by 1 inch in cross section. Test coupons 4 inches long were 

cut from the ends of each strip. The stripe were then ripped to the proper size 

of stiffener and glued to the concave face of the curved plate with cold-setting resorcinol glue. 

Static-bending and compression coupons were cut from the outside edge of each 

panel while trimming to finished size. 

All plates were made as shown in the sketches of figure 1. The two curved end 

pieces were notched to a width and depth sufficient to receive the axial 

stiffener. After the stiffener had been glued to the panel, the panel was 

nail-glued with a cold-setting resorcinol glue to the end pieces and conditioned in an atmosphere at a temperature of 75° F. and a relative humidity of 64 per­ cent. Prior to testing, the two curved ends of the plate were carefully 

machined to Insure flatness of the plywood and to insure equal length of Ply­ wood and axial stiffener. The side rails were placed on the plate at the time 

of test, thus providing simply supported edges. 

Methods of Test 

Compression testa of the plywood plates were made in a 100,000-pound hydraulic testing machine. The specimen was centered on a heavy steel plate supported 

on a 9-inch knife edge (fig. 2). Thus the lower bearing was in effect a 

4Forest Products Laboratory Report No 1562. "Longitudinally Stiffened Thinwalled Plywood Cylinders in Axial Compression," by E. W. Kuenzi. 1947. 

Rept. No. 1567 -2-
spherical head with negligible friction in a direction perpendicular to the 

knife edge. It was thought that this type of head would not retard the for­ mation of buckles when the buckling load was reached. The top of the specimen rested squarely against the upper head of the testing machine. 

A small initial load was put on the plywood plate and the plate carefully

centered and adjusted to obtain uniform bearing on the ends. The load was 

then increased to about one-fourth or one-half of the estimated buckling load, and the lateral deflection measured by a dial reading to 0.001 inch. If the 

dial reading changed appreciably, the load was reduced and the plate recentered 

and readjusted. The load was then increased and lateral deflections again observed. Corrections were made as necessary until the applied load produced negligible deflection (of the order of 0 to 0.002 inch). Thus bending due to eccentricity of loading was reduced to a minimum amount. 

Two metalectric strain gages were attached at the center of the panel, the 

inner gage being glued to the stiffener and the outer gage glued directly op­ posite on the convex face of the plate. Strain and lateral deflection readings were taken at regular increments of load, 

In all tests the plywood buckling was rapid and was accompanied by a drop in 

load. Movement of the testing machine heed was stopped when the plywood

buckled and the buckle pattern observed. If the stiffener had not buckled, 

loading was resumed until the stiffener buckled or failed. In several in­ stances, the load required to buckle or cause failure of the stiffener was 

greater than the load required to buckle the plywood. The highest load sup­ ported by the specimen was recorded as the maximum load. 

The static bending and compression coupons of plywood were tested in accordance 

with the A.S.T.M. tentative methods.5 The properties of the stiffeners were obtained from 1- by 4-inch coupons using the same apparatus and techniques applied to the plywood compression coupons. 

Computation of Results 

The theoretical buckling stress of the cylinder comparable to each curved 

plate was computed by the formula3 

where: 

p buckling stress, pounds per square inch. k = a constant, obtained as indicated below. 

EL= modulus of elasticity parallel to the grain of the wood in the 

veneers of the plywood, pounds per square inch. 

5"Tentative Methods of Testing Plywood, Veneer, and Other Wood and Wood-base 

Materials." A.S.T.M. Designation D805-4ST. 

Rept. No. 1567 -3-
h = thickness of plywood, inches. 

r = mean radius of curvature, inches. 

The observed buckling stress of the plywood plate was calculated by dividing

the load carried by the plywood at buckling by the loaded area. The load carried 

by an axial stiffener when the plywood buckled was calculated from the size, 

modulus of elasticity, and average strain of the stiffener. The total load 

supported by the stiffened plate is the sum of the load supported by the ply­ wood and that supported by the stiffener. 

The following notation applies in addition to that above: 

E1 = modulus of elasticity in bending of the plywood in the axial direction of the curved plate, pounds per square inch. 

E2 = modulus of elasticity in bending of the plywood in the cir­ cumferential direction of the curved plate, pounds per 

square inch. 

Eb = modulus of elasticity in compression of the plywood in the axial direction of the curved plate, pounds per square inch. 

Ea = modulus of elasticity in compression of the plywood in the cir­ cumferential direction of the curved plate, pounds per 

square inch. 

ET = modulus of elasticity tangential to the annual growth rings 

of the wood in the veneers of the plywood, pounds per 

square inch. 

EL can be calculated from the relations 

The ratio ET or (C) is taken as 0.050 for yellow birch and 0.045 for yellow- EL 

poplar, based on recent values obtained at the Forest Products Laboratory. 

Theoretical values for the buckling constant, can be obtained by entering the curve of figure 3 with the proper 

in figure 4. 

ratio. The value of 

is plotted against the parameter The value of k can 

then be determined from the above relationships. These curves are a reproduction of those used in a previous report.3 

Rept. No. 1567 -4-
Presentation and Analysis of Data 

Table 1 presents the data used in determining the buckling stresses of the 

plywood plates. The plates are arranged in 13 groups of six plates each, with 

the plates of each group being of similar dimensions and plywood construction, 

The first plate listed in a group was tested without a stiffener, the second 

with a light stiffener, and the others with increasingly heavier stiffeners. 

Examination of the observed buckling stresses of the plates within a group

showed no tendency of an increase in critical stress due to the addition of 

the stiffeners. Variation between the buckling stresses within a group of 

plates can be attributed largely to (1) initial imperfections in the plates, 

(2) dimensional and physical differences between the minor specimens and the 

plates, and (3) inevitable experimental errors. 

A curved plate may be considered as a segment of a cylinder, and it would seem 

likely that a relationship might exist between their critical stresses. In 

figure 5 the observed buckling stress of each plate is plotted against the 

theoretical buckling stress, corrected for length, of a comparable cylinder. 

This figure shows that the observed values for the plates are only slightly 

greater, on the average, than those computed for comparable cylinders. The 

average observed stresses for the plates of groups "d" and "m" are about 35 

percent greater than those for comparable cylinders. 

If the length and/or width of a plate is sufficiently reduced to prevent the 

formation of a full-size natural buckle, it would seem that the critical 

stress would be increased. The effect of a decrease in length upon the criti­ cal stress for a complete cylinder is illustrated in figure 4. The stiffeners of a stiffened cylinder or of a stiffened curved plate and the restraints 

at the edges of a curved plate impose a maximum limitation on the circumfer­ ential size of the buckle formed and therefore may be expected to increase 

the critical stress. The width-to-radius ratios of the plates tested for 

this report varied from 0.25 to 1.0; thus they were segments of comparable

cylinders varying from 0.04 to 0.16 of the complete cylinders. In spite of 

this fact, as has been previously noted, the average observed critical values 

for the plates were found to be only slightly higher than those computed for 

the cylinders, indicating little increase due to the width limitation. This 

failure in obtaining a consistent increase in critical stress over that of a 

comparable cylinder may be understood from the theory developed in Forest 

Products Laboratory Report No, 1322-A,6 upon which the following discussion is based, 

The buckling phenomenon in a cylinder subjected to axial compression is 

greatly influenced by the presence of initial imperfections in the shape or 

material of the cylinder and upon the circumferential width of the buckle. 

Such initial imperfections are likewise found in curved plates, which may be 

considered as sections of cylinders. The width of the buckle might be limited 

by sufficiently reducing the width of the curved plate. Figure 6, which 18 

a reproduction of figures 10 to 13, inclusive, of Report No. 1322-A, shows 

6March, H. W. , "Buckling of Long, Thin Plywood Cylinders in Axial Compression (Mathematical Treatment) ," 1943. 

Rept. No. 1567 -5-
the relation between the compressive stress, p, and 

to the amplitude of the buckle. A family of curves 

is drawn for four values of initial buckle amplitude, 

for varying values of The value of 

which is proportional 

equal to where h is the plywood thickness, r is the radius of curvature, 

and b is the width of the buckle. The critical stress associated with each 

value of is the minimum value of the stress indicated by the corresponding 

curve. 

It Will be noted from the curves that the minimum value of pr/ELh, regardless 

of the value of is 0.095 at a value of of 0.350. This indicates that 

the width of the natural buckle of a cylinder of plywood, of the construction 

for which the curves were drawn, can be obtained from the relationship = 0.350 

If the full width of this natural buckle is not permitted to form, 

either by the use of heavy axial stiffeners or reduced panel width, the criti­ cal stress may be associated with 

increase in 

some higher value of and may, therefore, be increased. For low values of it may be noted that there is a sharp 

values 

increase in critical stress with an 

remains about the 

but that for large values 

of such as 0.5, the critical stress same for different 

of 

Since the addition of axial stiffeners to the panels tested for this report

did 

forms, because 

not greatly increase the buckling stress, it is evident that the values 

of were large. 

Although the above discussion is based on 

This is to be expected, particularly for thin cylindrical 

is the ratio of the amplitude of the original buckle to the thickness. curves for three-ply 

(1:2:1) plywood with axial face grain, a similar argument can be made for 

other constructions. 

It would appear, from the above, that the buckling stress could be increased 

by the addition of closely spaced heavy stiffeners, and tests on such stif­ fened plywood cylinders have been made.4 These tests show that an increase 

in the buckling stress of the plywood is obtained from such stiffening if the 

stiffeners are very closely spaced. 

Previous work on the buckling of unstiffened curved plywood plates in compression7 indicated a slight increase of the critical stress when the width was decreased to about one-half radian. These plates were of smaller curvature 

than those tested for this report, and it is therefore probable that the 

initial deflections, were smaller. 

7Forest Products Laboratory Report No. 1508, "Buckling of Thin, Curved, 

Plywood Plates in Axial Compression," by E. W, Kuenzi, 1944. 

Rept. No. 1567 -6-
Conclusions 

The following conclusions may be drawn from the results of the tests on thin 

curved plywood plates simply supported at the edges and loaded in axial com­ pression: 

(1) An axial stiffener does not increase the buckling stress of the plywood.

The stiffener carries additional load and, if stiff enough, breaks up the 

buckle pattern. 

(2) The buckling stress is approximately equal to or slightly greater than 

that of a thin-walled plywood cylinder, and may be calculated as such. 

(3) If local imperfections could be reduced and the curved plywood plates

made more perfectly, increases in critical stress might be attained by the 

addition of a stiffener that would reduce the width of the buckle. The plates 

tested were carefully made, however, and are thought to be as perfect as those made commercially; yet no appreciable increases were obtained. 

Rept. No. 1567 -7-

Figure 1.--Sketch showing the construction and dimensions of an 

axially stiffened curved plywood panel to be tested in axial 

compression. Unstiffened panel similar, but without stiffener. 

Z M 76606 F 
Figure 2.--Axial-compression test of a curved plywood panel having 

axial stiffener. 

Z M 76776 F 


Figure 5.--A comparison of the observed buckling stress of a curved 

plywood plate with the theoretical buckling stress of a comparable 

plywood cylinder. Plates are either unstiffened or have an axial 

stiffener. Cylinder stress calculated on basis of a plywood 

cylinder having the same length, thickness, radius of curvature, 

and mechanical properties as the plywood plate. Values are 

plotted from the data of the table 1. 

Z M 76604 F